math part2

MATHEMATICS: MATHEMATICS as a Teachable Moment P Meaning P Choice P Diversity P Trust P Time 1
Create Meaning : Student Projects Student Written Problems and Solutions Sports / Pets / Cooking Date / Special Days / Season / Weather Place (Home / Community / School) Games Discussion in pairs, small groups and as a class Create Meaning 2
Give Choices : Choices provide meaning through a sense of: Give Choices 3
Value Diversity: Value Diversity Diversity should be treated as a positive factor in the classroom. We need to: 4
Create a Climate of Trust: Create a Climate of Trust 5
Ensure There is Adequate Time: 6 Ensure There is Adequate Time
Review of Silent Mouthing: Review of Silent Mouthing Use the “silent mouthing technique: When students make errors give them hints, suggest that they are close, acknowledge that they are a step ahead or say, “That is the answer to a different question.” Student Feedback to give: to give: Slower processors and complex thinkers the time they need to do the question. ä ä 7
Review of Place Value: Review of Place Value Place value should be taught at least once a week but preferably a place value connection should be made almost every day. The connections to algebraic thinking should be made (collecting like terms) as this will pay off when doing operations with fractions and algebraic expressions. 8
Organization of the CURRICULUM: Organization of the CURRICULUM All four strands (ŸNumber Sense, ŸSpatial Sense, ŸProbability and Data Sense and ŸPattern and Relationship Sense) should be covered every month (every week in Primary). Problem solving often embeds three of the strands depending on whether the problem has a focus on spatial relationships or data relationships. It is usually preferable to introduce a new topic through a problem. The Japanese teachers use this technique effectively. 9
Making Meaning with the WEEKLY GRAPH: Making Meaning with the WEEKLY GRAPH 10
Slide11: Watch for the “big ideas” in the video. What teaching techniques are effective? What Mathematical concepts are covered? 11
Slide12: 12
Slide13: NEW Strategies for OLD Ideas Intermediate Students Where do we find the time to teach this way? If students are taught this way, how will they do on the FSA tests? 13
Slide14: Multi-step Division and Decimal Fractions The first few times multi-step division is taught it should be done as a whole class. The errors made should be used as opportunities to investigate conceptual understanding. Placement of the decimal in the quotient should be done by asking, “Where does it make sense to put the decimal so that the answer makes sense?” 14
Slide15: Process for Teaching 1 ÷ 9 If possible, do multi-step division on grid paper (cm graph paper works well). If grid paper is not available, use lined paper turned sideways so that the lines become grids for keeping the numerals in the correct position. 15
Slide16: . 1 ÷ 9 16
Slide17: Many algorithms are culture specific time savers that create accuracy. 17
Slide18: Many algorithms are culture specific time savers that create accuracy. In the middle ages we used a box or window method. 18
Slide19: Algorithms in the 21st Century Algorithms should be developed through discussion with learners because the purpose of teaching algorithms is to develop understanding. The focus should be on accuracy, then on efficiency. The most efficient algorithm today is always based on today’s technology. The most efficient algorithm today is the calculator or the computer but we do need to understand the underlying concept or we don’t know if the answer makes sense. 19
Slide20: FRACTIONS are RICH in PATTERNS Working at your table or in your group, assign different members of the group to find the decimal fraction for: 20
Slide21: Common Fractions Simplest Form Decimal Equivalent Percentage Equivalent On your “Memorable Fractions” sheet please write in all the fractions studied in the video. Include some of the equivalent fractions for these. For example: (the first fraction illustrated in the video) There were three other fractions in the problem. There was one fraction from the graph. 21
Slide22: Have the students draw a bar graph of the results. Do not give students criteria for creating a good graph. Discuss the results and focus on the fact that graphs are supposed to give you a lot of information at a glance. This means that the graph should be neat, have a title and a legend (if necessary). In the end, the class will have developed assessment criteria from a meaningful context by having students notice what makes a graph a good communication tool. Self-evaluation is often the most effective. 22
Slide23: Have students discuss (write) what they know about the class by analyzing the data (graph). Use the think/pair/share method to create discussion, then share as a group (valuing diversity, creating trust and developing meaning through choice). Can they think of any questions or extensions? Use these for further research. 23
Slide24: Collect data. Decide which fractions (decimals and percents) you wish to study. If you are worried about coloring in the hundreds squares for a tricky fraction, leave this part until the next day and try it yourself. Enter the fractions on the Memorable Fraction sheet. Draw a circle graph of the data. Review the criteria. 24
Slide25: Can be rich in CURRICULUM Connections If the number of voters in the class is: 12 15 18 20 24 30 36 or Do the following: 25
Slide26: Find the ones. 5 This principle was used in the video to make equivalent fractions – in particular: PRINCIPLE of ONE 26
Slide27: PRINCIPLE of EQUIVALENCE Throughout the video and on the “Memorable Fractions” sheet, the students have been making equivalent fractions and have learned that every fraction can be expressed as an infinite number of common fractions, exactly one decimal fraction and one percentage fraction. It can also be expressed as a ratio. 27
Slide28: PRINCIPLE of BALANCE In the video one student noticed that when equivalent fractions are generated, both the numerator and denominator have to be multiplied by the same number. This is also an example of the Principle of One as: 28
Slide29: Please solve the following: Don’t forget to show your steps. 2x + 5 = 31 2x + 5 – 5 = 31 – 5 2x = 26 x = 13 29
Slide30: PRINCIPLE of ZERO This step is necessary for equation solving and is the only principle that is not generated in doing the “Weekly Graph”. It should have been generated much earlier in the primary grades when doing the “How Many Ways Can You Make a Number” activity during Calendar Time. 30
Slide31: How Many Ways? 31
Slide32: How Many Ways? 32
Slide33: Mark Criteria Mark Criteria P Where any sentence contains the Addition operation Where any sentence contains the Subtraction operation Where any sentence contains the Multiplication operation Where any sentence contains the Division operation Where any sentence contains more than two terms (e.g. 2 x 3 + 5 = 10) Where any sentence contains more than two operations (e.g. 2 x 3 + 4 = 10) Where any sentence contains a number more than the goal number (in this case 10) Where any sentence contains a number substantially greater than the goal number (in this case 50 or 100) Where any group of sentences shows evidence of a pattern (e.g. 1 + 9, 2 + 8, 3 + 7) Where any sentence shows knowledge of the power of zero (e.g. 6 – 6 + 10 = 10 or 10 + 0 = 10) Where any sentence uses doubling and halving to generate new questions (e.g. 4 x 6 = 24, 2 x 12 = 24, 1 x 24 = 24) Where any sentence shows knowledge of the power of one (e.g. 6 ÷ 6 + 9 = 10 or 10 x 1 = 10) Where any sentence shows knowledge of the commutative principle (e.g. 6 + 4 = 10 and 4 + 6 = 10) Where any sentence shows knowledge of the number Note: this applies only for numbers greater than 10, such as 24. In upper intermediate grades, award marks for exponential notation also. (e.g. 20 + 4 = 24 and 2 x 10 + 4 = 24) Where any sentence contains brackets, such as: (3 + 2) + (3 + 2) + (3 + 2) + (3 + 2) + 4 = 24 Where any sentence contains exponents, square roots, factorials, or fractions. Note: there should be no expectation of the demonstration of exponents, square roots or factorials before grade six, but their use should be acknowledged and rewarded where a student chooses to employ such operations in earlier grades. How Many Different Ways Can You Make a Number? 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 33
Slide34: PRINCIPLES of EQUATION SOLVING Principle of Zero Principle of One Principle of Equivalence Principle of Balance 34
Slide35: to 35
Slide36: PRINCIPLE of ONE 36
Slide37: to Equivalence is used in all facets of mathematics. Balance is used in equation solving as well as multiplication and division of rational expressions. The Principle of Zero is extensively in simplifying rational expressions. 37
Slide38: Probability can be introduced during the “Weekly Graph” process. Probability was introduced in the first session when playing “hangman” which is an activity students love to play. Probability sense is an important skill we use in everyday life. 38
Slide39: In your group, have one person shuffle the red deck (cards numbered 1 to 10) and a different person shuffle the blue deck. Ten-Frame Probability 39
Slide40: Was the most common prediction a 7? Ten-Frame Probability Turn over the two decks and find all the combinations that equal 7. 40
Slide41: NEW Strategies for OLD Ideas Intermediate Students Which ILOs were covered in the activity? What are some connected or follow-up activities that you could use? 41
Slide42: to Introducing the ten-frame cards this way allows grade four to eight students to look at numbers in a new way and learn to add visually without counting. The games shown in the video are called “Solitaire 10” and “Concentration 10”. Some students in intermediate grades have difficulty adding, and this is a new way to learn an old concept of making tens. 42
Slide43: “ALL THE FACTS” Sheet to 43
Slide44: SUBTRACTION FACTS to 44
Slide45: All of the fractions generated in the video were for ‘what you would expect to get’. This is called the “Expected Probability”. What we are really interested in is the “Experimental Probability”. The next step is to have each pair or students do 100 trials each and compare the Expected Probability to the Experimental Probability. The difference explains why people gamble. 45
Slide46: If each student in the class does 100 trials and then the data is put on a spreadsheet, it is clear that while some students will win if they pick their favourite number, others will lose. However, the experimental results for the whole class will usually mirror the expected probability. Government figures the odds, pays less than the expected probability, and makes lots of money. Gambling then is a tax on the under-educated, often the poor. 46
Slide47: 47
Slide48: Do the same activity with six-sided, ten-sided, or twelve-sided dice. Probability of getting a specific number or color of SmartiesTM or other candies on Halloween or Valentines Day. 48
Slide49: Take out the Decimal Fractions Project sheet. Enter all the fractions and decimals collected so far. Find the prime factorization of the denominator for each fraction (use fractions in their lowest terms only.) 49
Slide50: STUDENT FRACTION DECIMAL INVESTIGATION SHEET 9 = 3 x 3 5 = 5 10 = 2 x 5 so 4 = 2 x 2 = 50
Slide51: 0 1 Draw a line from 0 to 2. 2 51
Slide52: Sometimes it is important to have the number lines drawn vertically so that the student makes the connection to a thermometer. Then it is easy to introduce the idea of integers and negative integers in a natural context. 0 52
Slide53: MAKING MEMORIES In the last session the “Norman” story was introduced as a way to create a metaphor (based on scientific theory about the way we create memories) about how Norman learned to add 8 + 7 and other numbers by breaking the number up and using doubles. Other students were asked if they did the question in different ways and five responded. How can this story be used in a classroom when there is a student who yells out answers or interrupts with what he or she considers interesting comments? 53
Slide54: MAKING MEMORIES What have you mylenized over the course of the two videos? Please take 2 minutes of silence to write out a list. When the two minutes are up, the facilitator will ask you to share a strategy or concept you learned that you feel will be useful. This writing and then sharing helps “re-mylenize” your learning. 54
Slide55: Create a Class List with some or all of the following headings: 55
Slide56: Create Criteria for each Heading Example: Creates a CIRCLE Graph from Raw Data. 56
Slide57: Circle Graph Number of Siblings Zero siblings One sibling Two siblings Three siblings 27 students in the class told how many siblings they have. P P P Data Heading Legend Neatness P P P P 57
Slide58: EVALUATING Decimals / Fractions / Percentage Example for Multi-age Grade 6/7 (Grade 6 gets a 4 in the 3 category) 58
Slide59: Example for Multi-age Grade 6/7 (Grade 6 gets a 4 in the 3 category) Given a set of ordinary or special dice or a spinner, can create a data set and interpret both the expected and experimental probability. Creates a data set and interprets but makes some errors (not fundamental). Gets a good start and creates a data set but not both of expected and experimental. Barely gets started if at all, needs a lot of help. 59
Slide60: Teaching Classes There is some research that shows that students in multi-age classes demonstrate superior learning. This may result from the fact that the teacher knows she has to individualize more because of the spread of ability. In fact, this is true for all classes even when they are streamed. I have found it most effective when teaching a multi-age class to teach to the top grade and evaluate the lower grade at their own level. 60
Slide61: EVALUATION Work with someone at your table to create criteria for at least one of the Intended Learning Outcomes that you will be evaluating. Keep in mind that the creation of criteria is always a process of negotiation between you, the curriculum and your context (class and school). If you involve the students in creation of the criteria, they often create criteria that has a high standard of expectation for excellence. 61
Slide62: and Good Problem Solvers: Get started Get unstuck Persevere Can solve problems in more than one way Self-correct 62
Slide63: FACILITATING Problem Solving Use the think / pair / share method. Give problems that are multi-step and take note of student strategies. Record the strategies, slowly building up a list. Discuss the efficacy and efficiency of the various strategies that students use. 63
Slide64: FACILITATING Problem Solving Use model problems and have students write problems using the frame as a model. Encourage the use of mathematical vocabulary by giving bonus marks. Encourage the use of mathematical vocabulary by creating a word wall or a glossary in student workbooks. 64
Slide65: STRATEGIES for Getting Unstuck Look for a pattern Make a model Draw a diagram Create a table, chart or list Use logic Create a simpler related problem Work backwards Seek help from a peer, the internet, a book 65
Slide66: EVALUATING Problem Solving Example for Multi-age Grade 4/5 (grade 4 gets a 4 by achieving at the 3 level) 66
Slide67: IMPLEMENTATION Take the time to make a plan for implementation. What obstacles do you perceive? What help do you need? 67
Slide68: Fuson, Karen C., Kalchman, Mindy and Bransford, John D., Chapter 5, “Mathematical Understanding: An Introduction in How Students Learn Mathematics in the Classroom”, Ed. Donovan, Susanne and Bransford, John D., National Academies Press, Washington, D.C. 2005 Buschman, Larry E.E... Mythmatics” Teaching Children Mathematics, Vol.12, No.3, Oct. 2005, p136 –143 Calkins, Trevor “Mathematics as a Teachable Moment” Grades K-3, Power of Ten Educational Consulting Ltd, Victoria, B.C. 2004 Calkins, Trevor “Mathematics as a Teachable Moment” Grades 4 - 6, Power of Ten Educational Consulting Ltd, Victoria, B.C. 2004 Silver, Edward A and Cai, Jinfa. “Assessing Students’ Mathematical Problem Posing” Teaching Children Mathematics, Vol.12, No.3, Oct. 2005, p129 -135 Bibliography 68
Slide69: Slide presentation created by: Trevor Calkins Power of Ten Educational Consulting 809 Kimberley Place Victoria, B. C. V8X 4R2 www.poweroften.ca Power Point presentation constructed by: Karen Henderson P. O. Box 18 Shawnigan Lake, B. C. V0R 2W0 69

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